An ideal in a Noetherian ring is nilpotent if each element of the ideal is nilpotent.

Lemma 10.32.5. Let $R$ be a Noetherian ring. Let $I, J$ be ideals of $R$. Suppose $J \subset \sqrt{I}$. Then $J^ n \subset I$ for some $n$. In particular, in a Noetherian ring the notions of “locally nilpotent ideal” and “nilpotent ideal” coincide.

Proof. Say $J = (f_1, \ldots , f_ s)$. By assumption $f_ i^{d_ i} \in I$. Take $n = d_1 + d_2 + \ldots + d_ s + 1$. $\square$

Comment #851 by Bhargav Bhatt on

Suggested slogan: An ideal in a noetherian ring is nilpotent if each element of the ideal is nilpotent.

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